Two Seifert surfaces of links in $S^3$ are said to be twist equivalent if one can be obtained from the other, up to isotopy, by repeatedly performing operations consisting of cutting along an embedded arc, applying a full twist near one copy of the arc, and re-gluing. By using bridge spheres for their boundary links, we provide a new method of distinguishing twist equivalence classes of Seifert surfaces of any given genus. Given a Seifert surface $F$ of a link $L$, we show that the bridge numbers of the boundary links of Seifert surfaces twist equivalent to $F$ are uniformly bounded above by a constant depending only on $F$. By computing this constant in one simple case and applying a result of Pfeuti, we deduce that $b(L)\leq 2g_c(L)+|L|$ for any link $L$ in $S^3$, where $g_c(L)$ denotes the canonical genus of $L$. Various consequences of this inequality are discussed. We then apply the main theorem to show that there are infinitely many distinct twist equivalence classes represented by free, genus one Seifert surfaces, answering a question of Pfeuti in the negative.